Laser apparatus

ABSTRACT

A laser apparatus includes a resonator and an excitation source for gain section excitation. The resonator includes a gain section in which a population inversion is obtained by optical excitation or current injection, non-gain sections in which gain with respect to a laser oscillating light beam is not positive, and two reflection mirrors. The gain section has an optical path length approximately one half that of the resonator and placed centrally with respect to the optical axis of the resonator. The non-gain sections are arranged on opposite sides of the gain section. The two reflection mirrors are arranged further beyond the non-gain sections.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a laser apparatus and, morespecifically, to a mode-locked type laser apparatus.

2. Description of the Background Art

As the total capacity of transmission has been rapidly increasingrecently, in the field of optical fibers, increase of the transmissionrate per unit wavelength has been desired as an approach for practicalimplementation of ultra high speed/ultra large capacity opticaltransmission. For this purpose, in order to realize optical transmissionat a rate of exceeding 100 Gbps per unit wavelength in the near future,mode-locking semiconductor laser diodes (MLLD) have been studied. Themode-locked state includes two different types, that is, amplitudemodulation mode locking (AMML) and phase (frequency) modulation modelocking (FMML). In the AMML, all modes oscillate in-phase, and thereforea high energy optical pulse can be obtained. In the FMML, of theoscillating modes, some are oscillating out-of-phase, and therefore, ahigh energy optical pulse cannot be obtained. Therefore, laseroscillation of AMML is desired in order to obtain a high energy opticalpulse. In order to realize ultra high speed and ultra large capacityoptical transmission, no matter what repetition frequency is necessary,an AMML laser device that operates at that frequency is desired.

One example of the study of MLLD is disclosed in “All-Optical SignalProcessing with Mode-Locked Semiconductor Lasers”, Yokoyama et. al.,Proceedings of a symposium by The Institute of Electronics Informationand Communication Engineers. The MLLD described by Yokoyama et. al. isadapted to have a saturable absorption effect within the device, wherebybasic optical signal processing function as a coherent pulse lightsource, optical clock extraction, optical gate, as well as optical 3Ridentification reproduction (3R: retiming, reshaping, regenerating) andoptical time division demultiplexing (optical DEMUX) can be attained bya simple arrangement. In the MLLD technique by Yokoyama et. al.,mentioned above, however, a saturable absorption section is necessarywithin the laser cavity. Therefore, an operational instability, such ashysteresis of the light-current characteristic, is a serious problem forthe device. Further, the threshold current is high undesirably.

Therefore, an object of the present invention is to provide a laserapparatus not requiring the saturable absorption section and in whichmodes all oscillate in a mode-locked manner with constant phasedifferences regardless of the oscillating condition, that is, a laserapparatus that oscillates in AMML.

SUMMARY OF THE INVENTION

The above described object of the present invention is attained by thelaser apparatus by the present invention, which includes a resonator anda gain section excitation means wherein the resonator includes a gainsection in which population inversion is attained by at least one methodselected from the group consisting of optical excitation and currentinjection, non-gain sections in which gain with respect to the laseroscillating light beam is not positive, and two reflection mirrors. Thegain section is arranged at a central portion along the optical axis ofthe resonator, to have the optical path length approximately one halfthat of the resonator. The non-gain sections are arranged on both sidesof the gain section along the optical axis of the resonator. The tworeflection mirrors are arranged further outside of the non-gain sectionsalong the optical axis of the resonator. The gain section excitationmeans is for retaining the excited state of the gain section.

By the above described arrangement, it becomes possible to obtain AMMLoscillation even in a laser apparatus having such a resonator lengththat causes FMML oscillation when the inner portion as a whole of theresonator is used as the gain section.

In the above described invention, preferably, the gain section includesa semiconductor, and the non-gain section includes a dielectric.Alternatively, the gain section includes a semiconductor and thenon-gain section includes a semiconductor. By this arrangement, even ina semiconductor laser device, an arrangement that can attain AMMLoscillation can be realized easily.

In the above described invention, preferably, a non-gain sectionelectrode is provided, which performs carrier injection to the non-gainsection or application of a reverse bias to the non-gain sections. Inthis arrangement, it becomes possible to inject carriers to an opticalwave guide layer by using the non-gain section electrode and, as aresult, index of refraction within the resonator changes because of theplasma effect of free carriers, controlling substantial optical pathlength. Alternatively, by applying the reverse bias, the substantialoptical path length can be controlled. Accordingly, repetition frequencyof the mode locked optical pulse can be controlled.

In the above described invention, preferably, at least one of thereflection mirrors is a distributed Bragg reflector. By thisarrangement, it becomes possible to select the oscillation wavelength ofthe mode locked laser, and expansion of the oscillation spectrum can belimited.

In the above described invention, preferably, a distributed Braggreflector electrode is provided, for changing reflection spectrum bycurrent injection or reverse bias application to a periodic structure ofthe distributed Bragg reflector. When this arrangement is employed, itbecomes possible to change the reflection spectrum of the distributedBragg reflector and to change the wavelength of the mode locked opticalpulse.

The foregoing and other objects, features, aspects and advantages of thepresent invention will become more apparent from the following detaileddescription of the present invention when taken in conjunction with theaccompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic diagram of the apparatus in accordance with thefirst embodiment of the present invention.

FIG. 2 is a cross sectional view of the laser apparatus in accordancewith the second embodiment of the present invention.

FIG. 3 is a graph representing spatial distribution of potential andquasi-Fermi energy in an active section, in the second embodiment of thepresent invention.

FIG. 4 is a graph representing first order gain and dispersion spectra.

FIG. 5 is a graph representing a spectrum of the ratio d/l of a detuningterm and a locking coefficient, in accordance with the second embodimentof the present invention.

FIG. 6 is a graph representing a spectrum of locking coefficient l, inaccordance with the second embodiment of the present invention.

FIG. 7A is a graph representing a spectrum when a semiconductor laserhaving a resonator length of 300 μm is pulse-driven with a currentdensity of 41 kA/cm².

FIG. 7B is a graph representing spectra of three modes obtained ascalculation results.

FIG. 7C is a graph representing the result of an experiment with a laserhaving a resonator length of 600 μm.

FIG. 7D is a graph representing the results of calculation of the laserhaving the resonator length of 600 μm.

FIG. 8A is a graph showing the relation between the carrier density andspectral ranges where relations |d/l|<1 and l<0 are satisfied for a 300μm cavity.

FIG. 8B is a graph showing the relation between the carrier density andspectrum ranges where relations |d/l|<1 and l<0 are satisfied for a 400μm cavity.

FIG. 8C is a graph showing the relation between the carrier density andspectrum ranges where relations |d/l|<1 and l<0 are satisfied for a 500μm cavity.

FIG. 8D is a graph showing the relation between the carrier density andspectrum ranges where relations |d/l|<1 and l<0 are satisfied for a 600μm cavity.

FIG. 9A is a graph showing the relation between the carrier density andthe spectrum range where relations |d/l|<1 and l<0 are satisfied for a1083.6 μm cavity, in a heterogeneous excitation structure.

FIG. 9B is a graph showing the relation between the carrier density andthe spectrum range where relations |d/l|<1 and l<0 are satisfied for a1083.6 μm cavity, in a homogenous excitation structure.

FIGS. 10A to 10F are graphs representing carrier density dependency ofmode intervals, obtained as results of calculations for variousresonator lengths and resonator configurations.

FIG. 11 is a cross sectional view of the laser apparatus in accordancewith the third embodiment of the present invention.

FIG. 12 is a cross sectional view of the laser apparatus in accordancewith the fourth embodiment of the present invention.

FIG. 13A is a graph representing a reflection spectrum of a resonatorconsisting of a gain section (Table 3: A gain section) having a lengthof 1000 μm.

FIG. 13B is a graph representing a transmittance spectrum of a resonatorconsisting of a gain section (Table 3: A gain section) having a lengthof 1000 μm.

FIG. 14 is a graph representing a reflection spectrum of a resonatorconsisting of “transparent section (B) of 250 μm+gain section (A) of 500μm+transparent section (B) of 250 μm.”

FIG. 15 is a graph representing a reflection spectrum of a resonatorconsisting of “transparent section (B) of 270 μm+gain section (A) of 500μm+transparent section (B) of 250 μm.”

FIG. 16 is a graph representing a reflection spectrum of a resonatorconsisting of “DBR (Distributed Bragg Reflector) consisting of aperiodic structure of transparent section (B) and transparent section(C)+transparent section (B) of 250 μm+gain section (A) of 500μm+transparent section (B) of 250 μm”, where the number of pairs of theDBR is 40.

FIG. 17 is a graph representing a reflection spectrum of a resonatorconsisting of “DBR consisting of a periodic structure of transparentsection (B) and transparent section (C)+transparent section (B) of 250μm+gain section (A) of 500 μm+transparent section (B) of 250 μm”, wherethe number of pairs of the DBR is 500.

FIG. 18 is a graph representing a reflection spectrum of a resonatorconsisting of “DBR consisting of a periodic structure of transparentsection (B) and transparent section (C)+transparent section (B) of 250μm+gain section (A) of 500 μm+transparent section (B) of 250 μm”, wherethe number of pairs of the DBR is 1000.

FIG. 19 is a graph representing a reflection spectrum of a resonatorconsisting of “DBR consisting of a periodic structure of transparentsection (B) and transparent section (C)+transparent section (B) of 250μm+gain section (A) of 500 μm+transparent section (B) of 250 μm”, wherethe number of pairs of the DBR is 2000.

FIG. 20 is a graph representing a reflection spectrum of a resonatorconsisting of “DBR consisting of a periodic structure of transparentsection (B) and transparent section (C)+transparent section (B) of 250μm+gain section (A) of 500 μm+transparent section (B) of 250 μm”, wherethe number of pairs of the DBR is 5000.

FIG. 21 is a graph representing a reflection spectrum where the numberof pairs of the DBR is 2000 and the length of one transparent section is270 μm.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

(First Embodiment)

Referring to FIG. 1, the arrangement of the laser apparatus inaccordance with the first embodiment of the present invention isdescribed below. The laser apparatus includes a resonator 1 and anexcitation source 10 as a means for excitation of the gain section.Resonator 1 includes reflection mirrors 4 at both ends in the directionof the optical axis (left-right direction in FIG. 1), and at the portionbetween the reflection mirrors 4, there are a gain section 2 andnon-gain sections 3, as shown in FIG. 1. Here, “gain section” means asection at which population inversion is obtained by optical excitationor injection of current, and “non-gain section” means a section in whichthe gain with respect to the laser oscillating light beam is notpositive. More specifically, the non-gain section means a section thatis either transparent or slightly absorptive to the laser light. Aninterposed layer of air, for example, is also applicable to the non-gainsection.

As shown in FIG. 1, the central portion is the gain section 2 andopposing sides are the non-gain sections 3. Further, on the opposingsides thereof, two reflection mirrors are arranged. The length of gainsection 2 must be at least 8% of the length of resonator 1 along theoptical axis (hereinafter referred to as “resonator length”). Further,it should desirably be at most 50%. At least one of the reflectionmirrors is capable of partially transmitting the light beam. It isdesirable that reflection at an interface between the gain section 2 andthe non-gain section 3 is suppressed as much as possible.

For the case that the inner portion of the resonator is fulfilled with auniform gain section 2, oscillation starts when gain section 2 isexcited. In that case, oscillation occurs in the mode locked statewithout exception. As at least one of the reflection mirrors 4 canpartially transmit the light beam, mode locked optical pulses 16 areemerged from one of the reflection mirrors 4. Here, as described above,there are two different types of mode-locked state, that is, AMML andFMML, and the oscillation mentioned above occurs in either one ofmode-locked states.

If oscillation occurs in FMML with the gain section 2 occupying theentire inner portion of resonator 1, oscillation in AMML inevitablyoccurs by adapting the gain section 2 such that the length of the gainsection 2 is shortened to a half of the resonator length and theposition is at the center of the resonator 1. To the contrary ifoscillation in AMML occurs with the gain section 2 occupying an entireinner portion of the resonator 1, oscillation in FMML inevitably occursby adapting the gain section 2 such that the length of the gain section2 is shortened to a half of the resonator length and the position is atthe center of the resonator 1. Therefore, by selecting the arrangementof gain section 2, a laser apparatus that oscillates in AMML can berealized.

Therefore, the arrangement in accordance with the present invention isproposed as an arrangement to attain AMML oscillation, even in a laserapparatus having such a resonator length that causes FMML oscillation,when the entire inner portion of resonator 1 is used as the gain section2.

(Second Embodiment)

With reference to FIG. 2, the laser apparatus, that is the secondembodiment of the present invention will be described. The laserapparatus is formed basically following the arrangement of the firstembodiment, however, specifically it consists of semiconductormaterials. As shown in FIG. 2, a negative electrode 12 is formed on alower surface entirely along the resonator. On negative electrode 12, ann clad layer 14 is formed, and on n clad layer 14, a p clad layer 13 isformed. In gain section 2, the portion sandwiched between n clad layer14 and p clad layer 13 serves as an active section 15. In the gainsection 2, a positive electrode 11 is formed on p clad layer 13.Therefore, in this example, both the gain section 2 and the non-gainsections 3 consists of semiconductor materials. The non-gain sections 3,however, can be a dielectric material. The length of gain section 2 islimited to one half the length of resonator length and the gain sectionis arranged at the center of resonator 1.

When a first potential is applied to the positive electrode 11 and asecond potential lower than the first potential is applied to thenegative electrode 12 simultaneously, the gain section 2 attains to theexcited state, laser oscillation starts, and a mode locked optical pulse16 is generated.

Even when the laser apparatus has such a laser resonator length thatcauses FMML oscillation when the entire inner portion of the resonatorserves as the gain section, AMML oscillation can be attained if the gainsection 2 is arranged in the above described manner. When the resonatorlength is long, oscillation tends to be in FMML when the entire innerportion of the resonator is used as the gain section. Even in that case,however, AMML oscillation can be attained when the gain section 2 isarranged in the above described manner.

The reason why the arrangement of gain section 2 in the first and secondembodiments results in AMML oscillation is clarified by numericalanalysis of mode lock generation conditions, taking a quantum wellsemiconductor laser as an example. Therefore, a numerical formula modelused for the numerical analysis will be described, and the results ofnumerical analysis using the single quantum well laser as the model willalso be described.

Though the single quantum well semiconductor laser is used as the modelhere, the effect of the present invention can also be attained inmultiple quantum well semiconductor lasers. Further, the effect of thepresent invention can be attained by semiconductor lasers having doubleheterojunction structures.

(Numerical Analysis)

Here, the condition for mode lock generation in the quantum wellsemiconductor laser will be numerically analyzed. In order to find theconditions for mode lock generation, it is necessary to assume asituation where at least three adjacent modes oscillate simultaneously.Here, coupling mode equations for amplitude and phases, including up tothird order perturbation term of the electric field is used. Suchcoupling mode equations can be derived from Maxwell's equations, on theassumption that the respective mode amplitudes vary slowly. (W. E. Lamb,Jr., Phys. Rev., 134, A1429 (1964), M. Sargent, III, M. O Scully and W.E. Lamb, Jr., Laser Physics, (Addison-Wesley, Tokyo, 1974)).Polarization included in Maxwell's equations is determined by theelectronic state in the active section of the semiconductor laser ofinterest. In a semiconductor laser, the main source of polarization is 2dimensional (2D) free carrier transitions in the quantum well. In thesemiconductor laser, carriers are excited through current injection. Itis assumed that a quasi-equilibrium state is established in the excitedstate. Hence, the electron and hole densities in the quantum well aredefined and rate equations thereof can be derived. (H. Haug and S. W.Koch, Quantum Theory of the Optical and Electronic Properties ofSemiconductors, Third Edition, (World Scientific, New Jersey, 1998)).The rate equation and the coupling mode equations of the electric fieldamplitudes and phases described above determine the operation of thesemiconductor laser. Therefore, conditions for generating ML is alsodetermined by these expressions.

Particularly, conditions for AMML generation are described by two simpleinequality relations derived from the equations of motion of the phase.The inequality relations are formally the same as those for two-levelsystems derived by Lamb and Sargent.

In a laser, 2D free carriers are supplied by the current to the quantumwell. The supply process is calculated, by assuming the drift diffusionmodel, and solving simultaneously the continuity equation of3-dimensional Poisson's equation. In the quantum well section, a traptime of 3D carriers to 2D carriers and escape time of 2D carriers to 3Dcarriers are assumed, and thus the continuity equation of 3D carriersand rate equation of 2D carriers are combined. Thus net 2D carrierdensity is determined.

The specific single quantum well laser will be taken as the model, andconditions for AMML generation will be studied numerically on the baseof the formulation described above.

(A. Formulation)

(A. 1 Maxwell Wave Equation)

The starting point is Maxwell's wave equation. The following discussionwill be focussed only on TE (Transverse Electric) modes, because most ofsemiconductor lasers oscillate in this mode. The discussion can readilybe extended to TM (Transverse Magnetic) mode. From the Maxwell'sequations, the following wave equation for the electric laser field ∈within the resonator is derived. $\begin{matrix}{{\left\{ {{- \left( {\frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}}} \right)} + {\mu_{0}\sigma \frac{\partial}{\partial t}} + {\mu_{0}ɛ_{0}\frac{\partial^{2}}{\partial t^{2}}}} \right\} ɛ} = {{- \mu_{0}}\frac{\partial^{2}p}{\partial t^{2}}}} & (1)\end{matrix}$

where ∈₀ represents the dielectric constant of vacuum, μ₀ represents thepermeability of vacuum, and σ is a constant representing the amount ofelectromagnetic energy dissipating from the resonator to the outside. Pis a polarization self-consistent with the electric field ∈ of thelaser. In the equation (1), it is assumed that the field is homogenousin the x direction. The z direction is the axial direction of theresonator. The laser electric field oscillating in multi-mode can beexpanded by normal modes of the resonator, as follows, $\begin{matrix}{{ɛ\left( {y,z,t} \right)} = {{\frac{1}{2}{\sum\limits_{n}\quad {{E_{n}(t)}\exp \left\{ {- {i\left( {{v_{n}t} + \varphi_{n}} \right)}} \right\} {Y_{n}(y)}{Z_{n}(z)}}}} + {c.c.}}} & (2)\end{matrix}$

Here Y_(n) (y) is the unnormalized and dimensionless electric fielddistribution function in the y direction of a guided wave. When thedistribution of the refractive index of the laser structure (andtherefore, specific susceptibility χ (y)) is known, it can be obtainedby solving the equation (1), substituting P=∈_(0χ)(y)∈, and σ=0. Z_(n)(z) represents z dependency of the unnormalized normal mode, which is asfollows:

Z _(n)(z)=sin β_(n) z  (3)

where β_(n) is given by

β_(n) =nπ/L  (4)

Similarly, polarization P in the resonator can be represented by thefollowing equation. $\begin{matrix}{{P\left( {z,t} \right)} = {{\frac{1}{2}{\sum\limits_{n}\quad {{P_{n}(t)}\exp \left\{ {- {i\left( {{v_{n}t} + \varphi_{n}} \right)}} \right\} {Y_{n}(y)}{Z_{n}(z)}}}} + {c.c.}}} & (5)\end{matrix}$

In the equation (2), E_(n), Y_(n) and Z_(n) are considered as realwithout losing generality. Therefore, ∈ is real, whereas P is imaginary.The real component of P oscillates in the same phase as ∈, and theimaginary component is shifted in phase by 90°. By inputting equations(2) and (5) into the equation (1), projecting to Z_(n) (z) and byapproximation in which a term smaller than the quadratic differential ofthe electric field and polarization with respect to time is neglected,the following equations can be obtained. $\begin{matrix}{{{\overset{.}{E}}_{n} + {\frac{1}{2}\frac{v_{n}}{Q_{n}}E_{n}}} = {{- \frac{1}{2}}\frac{v_{n}}{ɛ_{o}}{{Im}\left( P_{n} \right)}}} & (6) \\{{v_{n} + {\overset{.}{\varphi}}_{n}} = {\Omega_{n} - {\frac{1}{2}\frac{v_{n}}{ɛ_{o}}E_{n}^{- 1}{{Re}\left( P_{n} \right)}}}} & (7)\end{matrix}$

In deriving equations (6) and (7), the wave number and the frequencyproper to the resonator are defined. Namely, $\begin{matrix}{K_{n}^{2} \equiv {{{- \frac{1}{Y_{n}}}\frac{^{2}Y_{n}}{y^{2}}} + \beta_{n}^{2}}} & (8)\end{matrix}$

 Ω_(n) =cK _(n)  (9)

Here, c represents the speed of light. As the amount representingdissipation of light, the following quality factor Q_(n) is introduced,in place of σ. $\begin{matrix}{\sigma = {ɛ_{0}\frac{v_{n}}{Q_{n}}}} & (10)\end{matrix}$

(A. 2 Equation of Light and Free Carriers in the Active Section)

Dipole moment associated with free carrier transition in thesemiconductor is dominate the polarization in equations (6) and (7). Asingle particle density matrix with respect to the state k of freecarriers is represented by the following equation. $\begin{matrix}{{{{{\rho_{k}(t)} = {\sum\limits_{\lambda^{\prime},\lambda}\quad {{\rho_{\lambda^{\prime},\lambda,k}(t)}{{\lambda^{\prime},k}}}}}\rangle}{\langle{\lambda,k}}}} & (11)\end{matrix}$

Here, c represents a conduction band and v represents a valence band.Each component of the density matrix follows the following equation ofmotion under a perfect k conservation.

{dot over (ρ)}_(cck)=ξ_(ck)−γ_(ck)ρ_(cck) +i ⁻¹∈(t)(d _(cvk)ρ_(vck) −d_(vck)ρ_(cvk))  (12)

{dot over (ρ)}_(vvk)=ξ_(vk)−γ_(vk)ρ_(vvk) −i ⁻¹∈(t)(d _(cvk)ρ_(vck) −d_(vck)ρ_(cvk))  (13)

{dot over (ρ)}_(cvk)=(γ+iω _(k))ρ_(cvk) −i ⁻¹∈(t)d_(cvk)(ρ_(cck)−ρ_(vvk))  (14)

Here, d_(cvk) represents matrix element of the dipole moment, ξ_(ck)(ξ_(vk)) represents the excitation rate of the carriers to the state kof the conduction band (valence band). Further, γ_(ck) and γ_(vk)represent damping rates from the state k of the conduction band (valenceband).

γ in equation (14) is given as $\begin{matrix}{\gamma = {{\frac{1}{2}\left( {\gamma_{ck} + \gamma_{vk}} \right)} + \gamma_{ph}}} & (15)\end{matrix}$

Here, γ_(ph) represents a phase relaxation constant, and ω_(k)represents energy between transition levels.

ω_(k)=ω_(ck)−ω_(vk)  (16)

(A. 3 Polarization)

Polarization represented by using the components of the density matrixand the matrix components of the dipole moment is as follows.$\begin{matrix}{P = {\frac{1}{V}{\sum\limits_{k}\quad \left( {{\rho_{cvk}d_{vck}} + {\rho_{vck}d_{cvk}}} \right)}}} & (17) \\{\quad {= {{\frac{1}{V}{\sum\limits_{k}\quad {\rho_{cvk}d_{vck}}}} + {c.c.}}}} & (18)\end{matrix}$

In equation (18), V represents volume of the active section of thelaser. Equating the corresponding complex conjugate components inequations (18) and (5) and projecting to Z_(n), we obtain the followingequation for the polarization component P_(n) of the mode n.$\begin{matrix}{{{P_{n}(t)} = {\frac{2^{i{({{v_{n}t} + \varphi_{n}})}}}{{VM}_{n}N_{n}}{\sum\limits_{k}\quad {\int_{- \infty}^{\infty}{{Y_{n}^{*}(y)}\quad {y}{\int_{0}^{L}{\rho_{cvk}d_{vck}{Z_{n}^{*}(z)}\quad {z}}}}}}}}{where}} & (19) \\{M_{n} = {\int_{- \infty}^{\infty}{{Y_{n}}^{2}\quad {y}}}} & (20) \\{N_{n} = {\int_{0}^{L}{{Z_{n}}^{2}\quad {z}}}} & (21)\end{matrix}$

By representing the density matrix component in equation (19) in termsof the electric field and substituting equation (19) to equations (7)and (6), we obtain equations of motion for the electric field amplitudesand the phases that are consistent with the electron system. Then, thenext step is to solve equations (12), (13) and (14). In these equations,the term including the electric field ∈ is considered to be aperturbation. The solution can be expanded by a series of electricfield, to the following form. Here, up to the third order perturbationterm will be solved.

ρ(t)=ρ⁽⁰⁾(t)+ρ⁽¹⁾(t)+ρ⁽²⁾(t)+ρ⁽³⁾(t)+ . . .  (22)

(A. 4 0th Order Term of Density Matrix)

Now, ρ⁽⁰⁾ (t) is the term not including the electric field. The termrepresents the initial state of the electron system when the interactionbetween electrons and fields starts. Integrating formally equation (12)with assumption of ∈=0 we obtain, $\begin{matrix}{\rho_{cck}^{(0)} = {\xi_{ck}\gamma_{ck}^{- 1}}} & (23)\end{matrix}$

This equation represents the probability of electrons occupying thestate k in the conduction band. As a quasi thermal equilibrium state isconsidered here, this equation can also be represented as$\begin{matrix}{\rho_{cck}^{(0)} = \frac{1}{{\exp \left\{ {\left( {{\hslash\omega}_{ck} - \mu_{c}} \right)/{kT}} \right\}} + 1}} & (24)\end{matrix}$

where μ_(c) represents quasi-Fermi energy in the conduction band, andthe value is determined so as to provide the number of carriers in theconduction band, when the equation (24) is summed up for every level.More specifically, the following equation holds. $\begin{matrix}{{n_{e}V} = {\sum\limits_{k}\quad \frac{1}{{\exp \left\{ {\left( {{\hslash\omega}_{ck} - \mu_{c}} \right)/{kT}} \right\}} + 1}}} & (25)\end{matrix}$

where n_(e) represents free electron density in the conduction band, andV represents the volume of the system.

Similarly, ρ⁽⁰⁾ _(vvk) obtained by integrating the equation (13) underthe condition of ∈=0 represents probability of electrons occupying thestate k in the valence band. Now, for the valence band, let us considernot the electrons but holes. We represent distribution function of theholes as follows. $\begin{matrix}\frac{1}{{\exp \left\{ {\left( {{\hslash\omega}_{vk} - \mu_{v}} \right)/{kT}} \right\}} + 1} & (a)\end{matrix}$

Then, the expression related to ρ⁽⁰⁾ _(vvk) and the hole density can begiven as follows. $\begin{matrix}{\rho_{vvk}^{(0)} = {1 - \frac{1}{{\exp \left\{ {\left( {{\hslash\omega}_{vk} - \mu_{v}} \right)/{kT}} \right\}} + 1}}} & (26) \\{{pV} = {\sum\limits_{k}\quad \frac{1}{{\exp \left\{ {\left( {{\hslash\omega}_{vk} - \mu_{v}} \right)/{kT}} \right\}} + 1}}} & (27)\end{matrix}$

Here, p represents hole density in the valence band, and μ_(v)represents quasi-Fermi energy in the valence band.

(A. 5 First Order Term of Density Matrix)

By substituting equations (24) and (26) for (ρ⁽⁰⁾ _(cck)−ρ⁽⁰⁾ _(vvk)) inequation (14) and integrating the equation in consideration of theequation (2), the following expression related to ρ⁽¹⁾ _(cvk) can beobtained, under rotational wave approximation. $\begin{matrix}\begin{matrix}{\rho_{cvk}^{(1)} = \quad {{- \frac{1}{2}}i\quad \hslash^{- 1}d_{cvk}{N_{k}\left( {z,t} \right)}{\sum\limits_{\sigma}\quad {{E_{\sigma}(t)}\exp \left\{ {- {i\left( {{v_{\sigma}t} + \varphi_{\sigma}} \right)}} \right\}}}}} \\{\quad {{Y_{\sigma}(y)}{Z_{\sigma}(z)}{D\left( {\omega_{k} - v_{\sigma}} \right)}}}\end{matrix} & (28)\end{matrix}$

where

N _(k)(z,t)≡ρ_(cck) ⁽⁰⁾−ρ_(vvk) ⁽⁰⁾  (29)

The right hand side depends on neither z nor t when a semiconductorlaser itself is considered. When the inner portion of the resonator isto be partially excited as will be discussed later, N_(k) depends on zand t. Further, there is the following relation.

D _(x)(Δω)≡1/(γ_(x)+iΔω), x=ck,vk, or no sign  (30)

When the equation (28) is to be derived, it is considered that

E _(σ),φ_(σ) ,N _(k)  (b)

hardly changes within the time range of 1/γ, and therefore this isplaced outside of integration (rate equation approximation).

(A. 6 Second Order Term of Density Matrix)

By substituting equation (28) into equation (12), using

ρ_(cvk)=ρ_(vck) ^(*)  (31)

d _(cv) =d _(vc) ^(*)  (32)

the following equation can be obtained. $\begin{matrix}\begin{matrix}{{\overset{.}{\rho}}_{cck} = \quad {\xi_{ck} - {\gamma_{ck}\rho_{cck}} - {\frac{1}{4}\left( \frac{d_{cvk}}{\hslash} \right)^{2}N_{k} \times}}} \\{\quad {\sum\limits_{\rho}\quad {\sum\limits_{\sigma}\quad {E_{\rho}E_{\sigma}Y_{\rho}^{*}Y_{\sigma}Z_{\rho}^{*}Z_{\sigma}{D\left( {\omega_{k} - v_{\sigma}} \right)} \times}}}} \\{\quad {{\exp \left\lbrack {i\left\{ {{\left( {v_{\rho} - v_{\sigma}} \right)t} + \varphi_{\rho} - \varphi_{\sigma}} \right\}} \right\rbrack} + {c.c.}}}\end{matrix} & (33)\end{matrix}$

By integrating this equation, we have $\begin{matrix}\begin{matrix}{\rho_{cck}^{(2)} = \quad {{- \frac{1}{4}}\left( \frac{d_{cvk}}{\hslash} \right)^{2}N_{k} \times {\sum\limits_{\rho}\quad {\sum\limits_{\sigma}\quad {E_{\rho}E_{\sigma}Y_{\rho}^{*}Y_{\sigma}Z_{\rho}^{*}Z_{\sigma}{D\left( {\omega_{k} - v_{\sigma}} \right)}}}}}} \\{\quad {{{D_{c}\left( {v_{\rho} - v_{\sigma}} \right)}{\exp \left\lbrack {i\left\{ {{\left( {v_{\rho} - v_{\sigma}} \right)t} + \varphi_{\rho} - \varphi_{\sigma}} \right\}} \right\rbrack}} + {c.c.}}}\end{matrix} & (34)\end{matrix}$

where it is assumed that mode amplitude and the phase hardly change in1/γ_(c). By substitution of γ_(cck)→γ_(vvk), the following equation isobtained. $\begin{matrix}{\rho_{vvk}^{(2)} = {- \rho_{cck}^{(2)}}} & (35)\end{matrix}$

From equations (34) and (35), we have the following relation.$\begin{matrix}\begin{matrix}{{\rho_{cck}^{(2)} - \rho_{vvk}^{(2)}} = \quad {{- \frac{1}{4}}\left( \frac{d_{cvk}}{\hslash} \right)^{2}N_{k} \times {\sum\limits_{\rho}\quad {\sum\limits_{\sigma}\quad {E_{\rho}E_{\sigma}Y_{\rho}^{*}Y_{\sigma}Z_{\rho}^{*}Z_{\sigma}}}}}} \\{\quad {{\exp \left\lbrack {i\left\{ {{\left( {v_{\rho} - v_{\sigma}} \right)t} + \varphi_{\rho} - \varphi_{\sigma}} \right\}} \right\rbrack} \times \left\{ {{D\left( {\omega_{k} - v_{\sigma}} \right)} +} \right.}} \\{{\quad \left. {D^{*}\left( {\omega_{k} - v_{\rho}} \right)} \right\}}\left\{ {{D_{c}\left( {v_{\rho} - v_{\sigma}} \right)} + {D_{v}\left( {v_{\rho} - v_{\sigma}} \right)}} \right\}}\end{matrix} & (36)\end{matrix}$

As can be seen from the equation (36), there appears a beat between themodes in the excitation distribution (“population pulsation”).

(A. 7 Third Order Term of Density Matrix)

By substituting the equations (36), (24) and (26) to equation (14), andsolving equation (14) the following equation results. $\begin{matrix}\begin{matrix}{\rho_{c\quad \upsilon \quad k}^{(3)} = \quad {\frac{\iota}{8}\left( \frac{d_{c\quad \upsilon \quad k}}{\hslash} \right)^{3}N_{k} \times}} \\{\quad {\sum\limits_{\mu}{\sum\limits_{\rho}{\sum\limits_{\sigma}{E_{\mu}E_{\rho}E_{\sigma}Y_{\mu}Y_{\rho}^{*}Y_{\sigma}Z_{\mu}Z_{\rho}^{*}Z_{\sigma}}}}}} \\{\quad {\exp \quad \left\{ {{{\iota \left( {{- \nu_{\mu}} + \nu_{\rho} - \nu_{\sigma}} \right)}t} + {\iota \left( {{- \varphi_{\mu}} + \varphi_{\rho} - \varphi_{\sigma}} \right)}} \right\} \times}} \\{\quad {{D\left( {\omega_{k} - \nu_{\mu} + \nu_{\rho} - \nu_{\sigma}} \right)}\left\{ {{D\left( {\omega_{k} - \nu_{\sigma}} \right)} + {D^{*}\left( {\omega_{k} - \nu_{\rho}} \right)}} \right\} \times}} \\{\quad \left\{ {{D_{c}\left( {\nu_{\rho} - \nu_{\sigma}} \right)} + {D_{\upsilon}\left( {\nu_{\rho} - \nu_{\sigma}} \right)}} \right\}}\end{matrix} & (37)\end{matrix}$

(A. 8 Mode Polarization)

By substituting equations (28) and (37) to equation (19), the expressionof mode polarization up to the third order can be obtained. Now, it isassumed that N_(k) has such a spatial dependency as follows.

 N _(k)(y,z)=N _(k) S ^(y)(y)S ^(z)(z)  (38)

By substituting the equation (28) to the equation (19), the first orderterm of mode polarization can be represented by $\begin{matrix}\begin{matrix}{{P_{n}^{(1)}(t)} = \quad {\frac{{- \iota}\quad \hslash^{- 1}}{{VM}_{n}N_{n}}{\sum\limits_{\sigma}{{E_{\sigma}(t)}^{\iota {\{{{{({\nu_{n} - \nu_{\sigma}})}t} + {({\varphi_{n} - \varphi_{\sigma}})}}\}}} \times}}}} \\{\quad {\sum\limits_{k}{{N_{k}\left( d_{c\quad \upsilon \quad k} \right)}^{2}{D\left( {\omega_{k} - \nu_{\sigma}} \right)}S_{n\quad \sigma}^{y}S_{n\quad \sigma}^{x}}}}\end{matrix} & (39)\end{matrix}$

where $\begin{matrix}{S_{n\quad \sigma}^{y} = {\frac{1}{M_{n}}{\int_{- \infty}^{\infty}{Y_{n}^{*}Y_{\sigma}S^{y}{y}}}}} & (40) \\{S_{n\quad \sigma}^{x} = {\frac{1}{N_{n}}{\int_{0}^{L}{Z_{n}^{*}Z_{\sigma}S^{z}{z}}}}} & (41)\end{matrix}$

In equation (39), if n≠σ, P⁽¹⁾ _(n)(t) has a component that oscillatesat a frequency corresponding to the mode interval. We assume that thepolarization and amplitude change slowly. (Fast oscillation component isincluded in the phase. This approximation is referred to as SVAapproximation (slowly-varying-amplitude approximation)). In accordancewith the assumption, terms with n≠σ are neglected. $\begin{matrix}{{P_{n}^{(1)}(t)} = {\frac{- {\iota\hslash}^{- 1}}{V}{E_{n}(t)} \times {\sum\limits_{k}{{N_{k}\left( d_{c\quad \upsilon \quad k} \right)}^{2}{D\left( {\omega_{k} - \nu_{n}} \right)}S_{n\quad n}^{y}S_{n\quad n}^{z}}}}} & (42)\end{matrix}$

As to the third order term, the equation (37) is substituted to theequation (19), and integration related to z is evaluated first.$\begin{matrix}{{Z_{n}^{*}Z_{\mu}Z_{\rho}^{*}Z_{\sigma}} = {\frac{1}{8}\left\lbrack {{\cos \left\{ {\left( {\beta_{n} - \beta_{\mu} + \beta_{\rho} - \beta_{\sigma}} \right)z} \right\}} + {\cos \left\{ {\left( {\beta_{n} - \beta_{\mu} - \beta_{\rho} + \beta_{\sigma}} \right)z} \right\}} + {\cos \left\{ {\left( {\beta_{n} + \beta_{\mu} - \beta_{\rho} - \beta_{\sigma}} \right)z} \right\}}} \right\rbrack}} & (43)\end{matrix}$

Further, also by SVA approximation, only those that satisfy thefollowing equation are left.

ν_(n)−ν_(μ)+ν_(ρ)−ν_(σ)≅0  (44)

Namely,

n=μ−ρ+σ  (45)

As a result, among the equation (43), only the following term remains.$\begin{matrix}{\frac{1}{8}\left\lbrack {1 + {\cos \left\{ {2\left( {\beta_{\sigma} - \beta_{\rho}} \right)z} \right\}} + {\cos \left\{ {2\left( {\beta_{\mu} - \beta_{\rho}} \right)z} \right\}}} \right\rbrack} & (46)\end{matrix}$

Therefore, the following equation is obtained, $\begin{matrix}\begin{matrix}{P_{n}^{(3)} = \quad {\frac{\iota}{32}\hslash^{- 3}{\sum\limits_{\mu}{\sum\limits_{\rho}{\sum\limits_{\sigma}{E_{\mu}E_{\rho}E_{\sigma}{\exp \left( {\iota \quad \Psi_{n\quad \mu \quad \rho \quad \sigma}} \right)} \times}}}}}} \\{\quad {S_{n\quad \mu \quad \rho \quad \sigma}^{y}S_{n\quad \mu \quad \rho \quad \sigma}^{z}\left\{ {{D_{c}\left( {\nu_{\rho} - \nu_{\sigma}} \right)} + {D_{\upsilon}\left( {\nu_{\rho} - \nu_{\sigma}} \right)}} \right\} \times}} \\{\quad {{{{\sum\limits_{k}}}}N_{k}d_{c\quad \upsilon \quad k}^{4}{D\left( {\omega_{k} - \nu_{\mu} + \nu_{\rho} - \nu_{\sigma}} \right)}D\left( {\left. {\omega_{k} - {\nu_{\sigma}}} \right) +} \right.}} \\{\quad \left. {D^{*}\left( {\omega_{k} - \nu_{\rho}} \right)} \right\}}\end{matrix} & (47)\end{matrix}$

where $\begin{matrix}{\Psi_{n\quad \mu \quad \rho \quad \sigma}\quad = \quad {{\left( {\nu_{n} - \nu_{\mu} + \nu_{\rho} - \nu_{\sigma}} \right)t} + \varphi_{n} - \varphi_{\mu} + \varphi_{\rho} - \varphi_{\sigma}}} & (48) \\{S_{n\quad \mu \quad \rho \quad \sigma}^{y}\quad = \quad {\frac{1}{M_{n}}{\int_{- \infty}^{\infty}{Y_{n}^{*}Y_{\mu}Y_{\rho}^{*}Y_{\sigma}S^{y}{y}}}}} & (49) \\{S_{n\quad \mu \quad \rho \quad \sigma}^{z}\quad = \quad {\frac{1}{N_{n}}{\int_{0}^{L}{\left\lbrack {1 + {\cos \quad \left\{ {2\left( {\beta_{\sigma} - \beta_{\rho}} \right)z} \right\}} + {\cos \quad \left\{ {2\left( {\beta_{\mu} - \beta_{\rho}} \right)z} \right\}}} \right\rbrack S^{z}{z}}}}} & (50)\end{matrix}$

By combining equations (42) and (47), we obtain $\begin{matrix}{{P_{n}(t)} = {{P_{n}^{(1)}(t)} + {P_{n}^{(3)}(t)}}} & (51)\end{matrix}$

By substituting the equation (51) to equations (6) and (7), the equationof motion related to the amplitude and phase of each mode desired can beobtained.

(A. 9 Coupling Mode Equation)

First, the equation of motion related to the amplitude is as follows.$\begin{matrix}{{\overset{.}{E}}_{n} = {{a_{n}E_{n}} - {\sum\limits_{\mu}{\sum\limits_{\rho}{\sum\limits_{\sigma}{E_{\mu}E_{\rho}E_{\sigma}\quad I\quad m\left\{ {\vartheta_{n\quad \mu \quad \rho \quad \sigma}\quad {\exp \left( {\iota \quad \Psi_{n\quad \mu \quad \rho \quad \sigma}} \right)}} \right\}}}}}}} & (52)\end{matrix}$

Here, the first term on the right hand side of equation (52) representslinear gain, and the second term represents nonlinear gain saturation.Similarly, the equation of motion related to the phase can be obtainedby the similar process. $\begin{matrix}{{v_{n} + {\overset{.}{\varphi}}_{n}} = {\Omega_{n} + \sigma_{n} - {\sum\limits_{\mu}{\sum\limits_{\rho}{\sum\limits_{\sigma}{E_{\mu}E_{\rho}E_{\sigma}E_{n}^{- 1}R\quad e\left\{ {\vartheta_{n\quad \mu \quad \rho \quad \sigma}{\exp \left( {i\quad \Psi_{n\quad \mu \quad \rho \quad \sigma}} \right)}} \right\}}}}}}} & (53)\end{matrix}$

Here, the second term on the right hand side is a first orderdispersion, and the third term is a nonlinear dispersion. Thecoefficients of equations (52) and (53) are as follows. First, theamounts representing the linear gain and dispersion are as follows.$\begin{matrix}{a_{n} = {{\frac{v}{2\quad ɛ_{0}\hslash \quad \gamma \quad V}{\sum\limits_{k}^{\quad}{{N_{k}\left( d_{c\quad u\quad k} \right)}^{2}{L\left( {\omega_{k} - v_{n}} \right)}S_{n\quad n}}}} - {\frac{1}{2}\frac{v}{Q_{n}}}}} & (54) \\{\sigma_{n} = {\frac{v}{2\quad ɛ_{0}\hslash \quad \gamma \quad V}{\sum\limits_{k}^{\quad}{{N_{k}\left( d_{c\quad u\quad k} \right)}^{2}\left\{ {\left( {\omega_{k} - v_{n}} \right)/\gamma} \right\} {L\left( {\omega_{k} - v_{n}} \right)}S_{n\quad n}}}}} & (55)\end{matrix}$

In equations (54) and (55), L (ω_(k)−ν_(n)) is a Lorentz function.$\begin{matrix}{{L\left( {\omega_{k} - v_{n}} \right)} = \frac{\gamma^{2}}{\gamma^{2} + \left( {\omega_{k} - v_{n}} \right)^{2}}} & (56)\end{matrix}$

The coefficient for the nonlinear term is represented by the followingequation. $\begin{matrix}{\vartheta_{n\quad \mu \quad \rho \quad \sigma} = {\frac{W}{64\hslash^{3}ɛ_{0}V}S_{n\quad \mu \quad \rho \quad \sigma}\left\{ {{D_{c}\left( {v_{\rho} - v_{\sigma}} \right)} + {D_{\sigma}\left( {v_{\rho} - v_{\sigma}} \right)}} \right\} \times {\sum\limits_{k}^{\quad}{{N_{k}\left( d_{c\quad u\quad k} \right)}^{4}{D\left( {\omega_{k} - v_{\mu} + v_{\rho} - v_{\sigma}} \right)}\left\{ {{D\left( {\omega_{k} - v_{\sigma}} \right)} + {D^{*}\left( {\omega_{k} - v_{\rho}} \right)}} \right\}}}}} & (57)\end{matrix}$

(A. 10 Three Modes Operation)

The conditions for ML generation and stability will be studied based onequations (52) and (53). For this purpose, it is necessary to considerat least three adjacent modes. The reason for this is that the mode lockrefers to a phenomenon in which phase relation between modes is keptconstant. The equations of motion of the amplitude and the phase ofthree modes can be given by the following equations. $\begin{matrix}{{\overset{.}{E}}_{1} = {{E_{1}\left( {a_{1} - {\sum\limits_{m = 1}^{3}{\theta_{1m}E_{m}^{2}}}} \right)} - {I\quad m\left\{ {\vartheta_{1232}{\exp \left( {{- i}\quad \Psi} \right)}} \right\} E_{2}^{2}E_{3}}}} & (58) \\{\left. {{\overset{.}{E}}_{2} = {{E_{2}\left( {a_{2} - {\sum\limits_{m = 1}^{3}{\theta_{2m}E_{m}^{2}}}} \right)} - {I\quad m\left\{ {\vartheta_{2123} + \vartheta_{2321}} \right){\exp \left( {i\quad \Psi} \right)}}}} \right\} E_{1}E_{2}E_{3}} & (59) \\{{\overset{.}{E}}_{3} = {{E_{3}\left( {a_{3} - {\sum\limits_{m = 1}^{3}{\theta_{3m}E_{m}^{2}}}} \right)} - {I\quad m\left\{ {\vartheta_{3212}{\exp \left( {{- i}\quad \Psi} \right)}} \right\} E_{2}^{2}E_{1}}}} & (60) \\{{v_{1} + {\overset{.}{\varphi}}_{1}} = {\Omega_{1} + \sigma_{1} - {\sum\limits_{m = 1}^{3}{\tau_{1m}E_{m}^{2}}} - {R\quad e\left\{ {\vartheta_{1232}{\exp \left( {{- i}\quad \Psi} \right)}} \right\} E_{2}^{2}{E_{3}/E_{1}}}}} & (61) \\\begin{matrix}{{v_{2} + {\overset{.}{\varphi}}_{2}} = \quad {\Omega_{2} + \sigma_{2} - {\sum\limits_{m = 1}^{3}{\tau_{2m}E_{m}^{2}}} -}} \\{\quad {R\quad e\left\{ {\left( {\vartheta_{2123} + \vartheta_{2321}} \right){\exp \left( {i\quad \Psi} \right)}} \right\} E_{1}E_{3}}}\end{matrix} & (62) \\\begin{matrix}{{v_{3} + {\overset{.}{\varphi}}_{3}} = \quad {\Omega_{3} + \sigma_{3} - {\sum\limits_{m = 1}^{3}{\tau_{3m}E_{m}^{2}}} -}} \\{\quad {R\quad e\left\{ {\vartheta_{3212}{\exp \left( {{- i}\quad \Psi} \right)}} \right\} E_{2}^{2}{E_{1}/E_{3}}}}\end{matrix} & (63)\end{matrix}$

where

 θ_(nm) =Im(∂_(nnmm)+∂_(nmmn))  (64)

τ_(nm) =Re(∂_(nnmm)+∂_(nmmn))  (65)

Ψ≡Ψ₂₁₂₃=(2ν₂−ν₁−ν₃)t+2φ₂−φ₁−φ₃  (66)

In the following, a situation is assumed, where mode 1 oscillates on thelower energy side and mode 3 oscillates on the higher energy side, withmode 2 being the center.

Let us focus the time change of the phase. By calculating the equation(62)×2−equation (63)−equation (61), the following equation results.

{dot over (Ψ)}=d+l _(s) sin Ψ+l _(c) cos Ψ  (67)

where detuning d is $\begin{matrix}{d = {{2\sigma_{2}} - \sigma_{1} - \sigma_{3} - {\sum\limits_{m = 1}^{3}{\left( {{2\tau_{2m}} - \tau_{1m} - \tau_{3m}} \right)E_{m}^{2}}}}} & (68)\end{matrix}$

and mode lock coefficients l_(s) and l_(c) are as follows.$\begin{matrix}{l_{s} = {I\quad m\left\{ {{2E_{1}{E_{3}\left( {\vartheta_{2123} + \vartheta_{2321}} \right)}} + {\left( {\frac{\vartheta_{1232}E_{3}}{E_{1}} + \frac{\vartheta_{3212}E_{1}}{E_{3}}} \right)E_{2}^{2}}} \right\}}} & (69) \\{l_{c} = {R\quad e\left\{ {{{- 2}E_{1}{E_{3}\left( {\vartheta_{2123} + \vartheta_{2321}} \right)}} + {\left( {\frac{\vartheta_{1232}E_{3}}{E_{1}} + \frac{\vartheta_{3212}E_{1}}{E_{3}}} \right)E_{2}^{2}}} \right\}}} & (70)\end{matrix}$

The equation (67) can further be simplified to

{dot over (Ψ)}=d+l sin(Ψ−Ψ₀)  (71)

where $\begin{matrix}{l = {l_{s}\left( {1 + \frac{l_{c}^{2}}{l_{s}^{2}}} \right)}^{1/2}} & (72)\end{matrix}$

 Ψ₀=−tan⁻¹(l _(c) /l _(s))  (73).

Equations (58), (59), (60) and (71) determine the three modes motion.Mode lock appears when the following relation is satisfied.

{dot over (Ψ)}=0  (74)

Namely, by the definition (48) of Ψ, there is the following relation

ν₂−ν₁=ν₃−ν₂≡Δ  (75)

and therefore, a relation is determined between phases φ₁, φ₂ and φ₃.

By assuming that E_(n) is not dependent on time, the equation (71) canbe considered to be independent of equations (58), (59) and (60).Further, in order for the equation (71) to have a stationary solution,the following is necessary. (Condition of mode lock generation).

|d|<|l|  (76)

The condition of stability of the state where

{dot over (Ψ)}=0  (c)

will be studied. Now, the solution of equation (74) is represented byΨ^((s)), and ∈ is assumed to be a slight variation of Ψ from the stablestate.

By substituting the following equation

Ψ=Ψ^((s))+∈  (77)

to the equation (71), and neglecting higher order terms of second orhigher of ∈, the following equation results.

{dot over (∈)}=∈lcos(Ψ^((s))−Ψ₀)  (78)

By integrating this equation,

∈=exp{tl cos(Ψ^((s))−Ψ₀)}  (79)

Accordingly, if the following relation is satisfied,

l cos(Ψ^((s))−Ψ₀)<0  (80)

then, ∈→0, when t→∞ (condition of mode lock stabilization). The objectof this section is to obtain quantitative evaluation of the condition(76) of mode lock generation and condition (80) of stabilization, forthe actual semiconductor laser.

First, the condition for mode lock stabilization will be further brokendown. When the equation (74) holds, the equation (71) has the followingsolutions. $\begin{matrix}{\Psi^{(s)} = {\Psi_{0} - {\sin^{- 1}\left( {d/l} \right)}}} & (81) \\{\quad {= {{\Psi_{0} \pm \pi} + {\sin^{- 1}\left( {d/l} \right)}}}} & (82)\end{matrix}$

When the equation (81) holds, the following equation results if the timeorigin is taken so that φ₁=φ₂.

φ₁=φ₂=−sin⁻¹(d/l)+φ₃  (84)

Now, where

sin⁻¹(d/l)→0  (85)

namely,

 d/l→0  (86)

then, three modes come to have the same phase, and mode lock is obtainedin which all the modes are superposed in phase. In this case, there isthe following relation

cos(Ψ^((s))−Ψ₀)>0

and therefore condition (80) will be represented as

l<0  (87)

and similarly,

l _(s)<0  (88)

This is the condition for amplitude mode lock generation.

On the other hand, when the condition (82) is satisfied, the phaserelation can be represented by

φ₁=φ₂=±π+sin⁻¹(d/l)+φ₃  (89)

In this case, if the equation (85) holds, then modes 1 and 2 oscillatein phase, while mode 3 oscillates of phase. Therefore, the signalintensity of the mode locked pulse becomes smaller. Here, as cos(Ψ^((s))−Ψ₀)<0, the condition (80) can be represented as l_(s)>0. Thisis the condition of FMML generation.

Until the condition (76) of mode locked generation and either one of thecondition (88) i.e. AMML and condition (l_(s)>0) i.e. FMML aresatisfied, the phase of each mode of the laser fluctuates. Once any ofthe conditions is satisfied, oscillation is maintained stably at thatpoint. Generally, in a laser having the Fabry-Perot structureoscillating in multimode, it is the case that the oscillation is eitherin AM or FM state. Therefore, the object here is to find the conditionof AM mode lock generation (that is, the condition to satisfy |d/l|<1,l<0).

(A. 11 Rate Equation of Carriers in Quantum Well)

The rate equation of carriers in the quantum well can be derived in thefollowing manner from the equation (33). By summing Eq (33) with respectto k and dividing the sum by the volume of the quantum well, thefollowing equation results. $\begin{matrix}\begin{matrix}{{\overset{.}{n}}_{2D} = \quad {\Xi_{c} - {\gamma_{c}n_{c}} - {\frac{1}{4\quad V}{\sum\limits_{k}{\left( \frac{{}_{}^{}{}_{c\quad u\quad k}^{}}{\hslash} \right)^{2}N_{k} \times}}}}} \\{\quad {\sum\limits_{\rho}{\sum\limits_{\sigma}{E_{\rho}E_{\sigma}Z_{\rho}^{*}Z_{\sigma}Y_{\rho}^{*}Y_{\sigma}{D\left( {\omega_{k} - v_{\sigma}} \right)} \times}}}} \\{\quad {{\exp \left\lbrack {i\left\{ {{\left( {v_{\rho} - v_{\sigma}} \right)t} + \varphi_{\rho} - \varphi_{\sigma}} \right\}} \right\rbrack} + {c.c.}}}\end{matrix} & (90)\end{matrix}$

The third term on the right hand side of the equation (90) representsdecay rates due to stimulated emissions. The term is obviously time- andspace-dependent. In actual calculations, however, the field and carrierdensity are assumed to be homogenous in the resonator and have valuesaveraged in time and space. Consequently, stimulated emission ratesbecome homogenous as well. Further, in this embodiment, we consider twocases only. One is that the gain region occupies whole resonator and theother is that the gain region is restricted in length to one half of theresonator and is placed in just the middle of the resonator. For bothcases, terms that include different indices vanish by the averagingprocedure because of the orthogonality of the function Z (z). As aresult, the fast time variation of the third term also disappears. Thisapproximation corresponds to the approximation (SVA approximation) thatthe amplitude hardly varies in the round trip time of light in theresonator. In a quantum well in which n_(2D) is defined, the electricfield of light can be considered almost constant, and therefore,averaging of the electric field of light in the y direction within thequantum well may be a good approximation. By rewriting the equation (90)using equation (38), the following equation is obtained. $\begin{matrix}{{{\overset{.}{n}}_{2D} = {\Xi_{c} - {\gamma_{c}n_{2D}} - {\frac{1}{2V\quad \gamma}{\sum\limits_{\rho}{S_{\rho}^{z}S_{\rho}^{y}E_{\rho}^{2}{\sum\limits_{k}{{N_{k}\left( \frac{d_{c\quad u\quad k}}{\hslash} \right)}^{2}{L\left( {\omega_{k} - v_{\rho}} \right)}}}}}}}}{w\quad h\quad e\quad r\quad e}} & (91) \\{S_{\rho}^{z} = {\frac{1}{L}{\int{{S^{z}(z)}Z_{\rho}^{2}d\quad z}}}} & (92) \\{S_{\rho}^{y} = {\frac{1}{L_{w\quad e\quad l\quad l}}{\int{{S^{y}(y)}Y_{\rho}^{2}d\quad y}}}} & (93)\end{matrix}$

In equation (91), Ξ_(c) represents the excitation rate of carriers tothe active section. This amount is determined, microscopically, bytransportation of three-dimensional carriers n_(3D).

(A. 12 Transportation of Three Dimensional Carriers n_(3D))

Transportation of three dimensional carrier n_(3D) is governed by thefollowing Poisson equation and the continuity equation.

div(∈gradφ)=−ρ(y)  (94)

{dot over (n)} _(3D) +divJ ^(c) =−G  (95)

In equation (94), ρ (y) represents spatial distribution of charges. Inequation (95), G is a term representing generation and annihilation ofcarriers that realize the thermal equilibrium. In the quantum wellsection, 3D carriers transported in accordance with the above equationswill be scattered to 2D states and confined. Conversely, 2D carriers arealso scattered to 3D carriers at a certain probability, escaping fromconfinement state. Calculation of these two scattering probabilities isitself a significant problem. Here, respective probabilities will berepresented by two times, that is, trap time τ_(trap) and escape timeτ_(escape). Therefore, equations (91) and (95) in the quantum wellsection will be $\begin{matrix}{{\overset{.}{n}}_{2D} = {{{- n_{2D}}/\tau_{e\quad s\quad c\quad a\quad p\quad e}} + {n_{3D}/\tau_{t\quad r\quad a\quad p}} - {\gamma_{c}n_{2D}} - {\frac{1}{2V\quad \gamma}{\sum\limits_{\rho}{S_{\rho}^{z}S_{\rho}^{y}E_{\rho}^{2}{\sum\limits_{k}{{N_{k}\left( \frac{d_{c\quad u\quad k}}{\hslash} \right)}^{2}{L\left( {\omega_{k} - v_{\rho}} \right)}}}}}}}} & (96) \\{{{\overset{.}{n}}_{3D} + {d\quad i\quad v\quad J^{c}}} = {{- G} + {n_{2D}/\tau_{e\quad s\quad c\quad a\quad p\quad e}} - {n_{3D}/\tau_{t\quad r\quad a\quad p}}}} & (97)\end{matrix}$

The current in accordance with the drift diffusion model can berepresented by

J ^(c) =−nμ ^(c)grad(−φ+V ^(c))−D ^(cgrad n)  (98)

In the equation (98) of current, D represents a diffusion coefficient,and it is assumed here that Einstein relation D=μkT/q is satisfied.Further, in the present embodiment, the charge of an electron isconsidered to be −1. Though expressions related to electrons have so farbeen given, similar expressions hold with respect to the holes.

In the following, equations (58) to (63) of motion of the amplitude andphase of three modes and the continuity equations (96) and (97) of theelectronic system as well as the slow variation of electric field (94)will be numerically solved simultaneously, so as to find conditions(87), (86) for stable generation of AMML.

(B. Calculation)

(B. 1 Model)

Laser Structure Assumed for Calculation

The laser assumed for calculation has an SCH structure of single quantumwell, consisting of an InGaAsP/InP heterojunction structure,lattice-matched with an InP substrate (see Table 1). It oscillates at awavelength in a 1.55 μm band. We use values of material constants suchas bandgap energies that appear in G. P. Agrawal and N. K. Dutta,Long-Wavelength Semiconductor Lasers, (Van Norstrand Reinhold Company,N.Y., 1986).

TABLE 1 Laser Structure Assumed for Calculation Quantum P clad pSCH pSCHwell nSCH nSCH n clad Layer thickness (nm) 100 60 30 6 30 60 100 Arseniccomposition  0 0.6431 0.6431 1 0.6431 0.6431  0 Donner concentration 5 ×10¹⁵ 5 × 10¹⁵ 5 × 10¹⁵ 5 × 10¹⁵ 1 × 10¹⁷ 5 × 10¹⁷ 1 × 10¹⁸ (cm⁻³)Acceptor 1 × 10¹⁸ 5 × 10¹⁷ 3 × 10¹⁷ 3 × 10¹⁷ 2 × 10¹⁷ 1 × 10¹⁷ 5 × 10¹⁶Concentration(cm⁻³)

Parameter Values Used for Calculation

Among the parameters having direct influence on the mode lock condition,phase relaxation constant γ is assumed to have a constant value 30(ps⁻¹) in this calculation. (W. W. Chow, A. Knorr, S. Hughes, A. Girndt,and S. W. Koch, IEEE J. Selected Topics in Quantum Electron., 3, 136(1997)). Carrier density, n_(2D), γ_(c) and so on vary with theoperation condition. The value γ_(c) is assumed to be given by thefollowing equation, using spontaneous emission coefficient B.

γ_(c) =B(n _(2D) p _(2D))^(1/2)  (d)

The value τ_(cap) used for the calculation was 0.1 ps. The valueτ_(escape) for the electrons and holes were 0.167 ps and 0.128 ps,respectively. These values were selected such that the spectra observedin the experiment could be reproduced by changing the applied voltage(see FIGS. 7A to 7D). The values are listed in Table 2 below.

TABLE 2 Parameter Values Used for Calculation τ_(cap)(ps) 0.1 τ_(esc)(electron) (ps) 0.167 τ_(esc) (hole) (ps) 0.128 γ (ps⁻¹) 30 naturalradiation 0.04 coefficient B (nm³/ps) internal loss (cm⁻¹) 20

(B. 2 Example of Typical Results of Calculation)

FIGS. 3 to 6 show examples of numerical solutions of the equationsystems (58) to (63), (96), (97) and (94) for the laser with a 300 μmlength when applied voltage is 1.1V. FIG. 3 represents spatialdistribution of potential and quasi-Fermi energy in the gain section.The carrier density in the quantum well at this time is (np)=5.25×10¹⁸cm⁻³. FIG. 4 shows spectra of first order gain and dispersion. FIG. 5represents a spectrum of the ratio d/l of the detuning term to thelocking coefficient. It is clearly seen that |d/l|<1 is attained near0.8007 eV. When the laser oscillates within the spectrum range of|d/l|<1, time variation of the amplitude and phase becomes small, andhence oscillation becomes steady. In the numerical solution, the laseroscillates at

ν₂=0.8007 eV  (e)

The amplitude near this energy can be considered constant. Therefore,using the amplitude at

ν₂=0.8007 eV  (f)

the spectrum of |d/l| is drawn, which is FIG. 5 (“decoupledapproximation”). FIG. 6 shows a spectrum of the locking coefficient ofl. This spectrum is also correct near 0.8007 eV quantatively. At theoscillation energy,

ν₂=0.8007 eV  (g)

the condition |d/l|<1 and l<0 is satisfied. More specifically, with thissolution (oscillation energy and FIGS. 3 to 6), AMML is generated.

The curves of FIGS. 5 and 6 obtained under “decoupled approximation” aremeaningful. The spectrum range satisfying |d/l|<1 is narrow, and hence,the assumption that the amplitude does not vary significantly in thatspectral range may be adequate. Further, the aim of FIG. 6 is to showthe range where l<0 and it is clear from the expression (69) thatwhether l is positive or negative does not depend on the amplitude.

FIG. 7B represents spectrum of three modes obtained as a result ofcalculation. FIG. 7A represents the spectrum when the semiconductorlaser having the resonator length of 300 μm was pulse-driven with thecurrent density of 41 kA/cm². As will be described later, a clear AMMLpulse was observed in the experiment. FIGS. 7C and 7D are spectrarepresenting the result of experiment (20.5 kA/cm²) of the laser havingthe resonator length of 600 μm and the result of calculation. The valueτ_(esc) of Table 2 was selected so that the central frequencies of FIGS.7A and 7C can be covered by selecting the applied voltage V_(B) incalculation.

(B. 3 Condition to obtain AMML: Condition for Resonator Length (ModeInterval) and Carrier Density)

FIGS. 8A to 8D show the spectral ranges satisfying relations |d/l|<1 andl<0 as functions of carrier density for lasers with cavity length of300, 400, 500 and 600 μm. The relation is obtained based on thesolutions of the equation systems (58) to (63), (96), (97) and (94) whenV_(B) is varied from 0.79V to 1.1V. When the resonator length is withinthe range of 300 to 500 μm, the spectrum range satisfying |d/l|<1 foreach carrier density is almost covered by the spectrum range where l<0,and hence AMML can be obtained. When the resonator length is 600 μm, thespectrum range satisfying |d/l|<1 is separated from the spectrum rangesatisfying l<0, when the carrier density increases. In that case,oscillation will be FMML.

In this manner, the condition to obtain AMML depends on the carrierdensity and the resonator length (mode interval). When the resonatorlength increases, it becomes more difficult to obtain AMML. When thecarrier density increases, it becomes more difficult to obtain AMML. Inother words, in order to generate AMML, the following condition must besatisfied.

2γγ_(c)−Δ²<0  (99)

This is an empirical rule obtained through simulations. Further, theinequality (99) is a necessary condition and not a sufficient condition.By numerical calculation, when the resonator length is further increasedto 1083.6 μm, only the FMML is obtained, as shown in FIG. 9B. However,for this cavity length, AMML can take place again, as shown in FIG. 9A,by restricting the excitation region to a part of middle of the cavity(resonator) as shown in FIG. 2. Excitation of the prescribed sectiononly of the resonator in this manner will be hereinafter referred to as“inhomogeneous excitation.”

Recovery of AMML described above by inhomogeneous excitation can beunderstood from the equation (69). The coefficients included in l_(s)are limited to Imθ₂₁₂₃, Imθ₂₃₂₁, Imθ₁₂₃₂, and Imθ₃₂₁₂. For thesesuffixes, the term obtained by integrating cosine terms vanishes in thecase of homogenous excitation, and therefore the equation (50) is alwayspositive. By contrast, when only the central portion (the section of onehalf the length of the resonator) of the resonator is excited and othersections are kept transparent (that is, N_(k) (y, z)=0 in thesesections), integrations of two cosine terms added to each other come tohave a negative value of which absolute value is larger than 1, andtherefore, the value of equation (50) is always negative. Therefore,when FMML occurs by homogenous excitation of the entire resonator of theFP laser, AMML can be obtained without fail when only the centralportion (the section one half the length of the resonator) of theresonator is excited.

In the description above, the section without gain is considered as atransparent section. The portion may have slight absorption (that is,N_(k) (y, z)≦0 in this section), and the effect is the same, in thesense that AMML is obtained.

What should be noted from FIGS. 8A and 8B for the case of the resonatorlength of 300 and 400 μm, in which the AMML condition can be satisfiedin a wide spectrum range is that the spectral range satisfying |d/l|<1becomes wider, as the carrier density (and hence amplitude) increases.This is because the mode lock coefficient l_(s) increases as theamplitude increases, while the detuning of the linear dispersion isindependent of the amplitude. It is possible that mode lockingoscillation occurs in the wide range, in that case, it is expected thatthe time width of the mode locked pulse gradually becomes shorter. Thisis observed in the experiment.

Further, the magnitude of |d/l| corresponds to the phase difference (84)between the modes, and therefore, it is expected that the phasedifference decreases as the carrier density increases. This contributeto improvement of extinction ratio of the mode locked optical pulse.

(B. 4 Mode Interval)

The carrier density dependency of mode interval obtained as a result ofcalculation is plotted for respective resonator lengths, in FIGS. 10A to10F. Generally, the mode interval increases as the carrier densityincreases. What is important is that the curve seems to have a peak. Forexample, when the resonator length is 300 μm, the peak is at 5.2×10¹⁸cm⁻³. At this peak position, it is expected that fluctuation (jitter) ofrepetition frequency of the beat pulse caused by fluctuation of thecarrier density is minimized.

(Third Embodiment)

By referring to FIG. 11, the arrangement of the laser apparatus inaccordance with the third embodiment of the present invention will bedescribed. The laser apparatus is formed basically following thearrangement of the second embodiment, except that a non-gain sectionelectrode 5 as an electrode for injecting carriers to the non-gainsection 3 is attached.

In the laser apparatus in accordance with the present embodiment, it ispossible to inject carriers to the optical waveguide layer, using thenon-gain section electrode 5. By the plasma effect of free carriersassociated with carrier injection, index of refraction in the resonator1 changes, and hence substantial optical path length can be controlled.

Accordingly, the repetition frequency of mode locked optical pulses 16can be controlled.

(Fourth Embodiment)

By referring to FIG. 12, the arrangement of the laser apparatus inaccordance with the fourth embodiment of the present invention will bedescribed. The laser apparatus is formed basically following thearrangement of the second embodiment, except that reflection mirrors 4are not simple mirrors but distributed Bragg reflectors 4 a, consistingof a diffraction grating. On the upper surface of the distributed Braggreflector, a distributed Bragg reflector electrode 6, that is anelectrode for injecting carriers to the diffraction grating, isprovided.

In the laser apparatus in accordance with the present embodiment, byselecting the repetition frequency of the diffraction grating 4 a of thedistributed Bragg reflector, it is possible to control the wavelength atwhich the mode locked optical pulse is obtained. Further, it is possibleto make narrower the emission spectrum of the laser apparatus as needed.Further, by applying a forward or reverse bias between the distributedBragg reflector electrode 6 and the negative electrode 12, thereflection spectrum of the diffraction grating can be changed, andtherefore, the wavelength at which the mode locked optical pulse isobtained can be changed as necessary.

The change of index of refraction by providing the non-gain sectionelectrode 5 in the non-gain section 3 including the semiconductor in thethird and fourth embodiments, the function of the diffraction grating inthe fourth embodiment, and the effect obtained by changing the index ofrefraction by carrier injection, by the provision of distributed Braggreflector electrode 6 in the diffraction grating in the fourthembodiment will be described in the following.

When the resonator length is long, it is effective to limit the gainsection to the central portion along the optical axis of the resonator,in order to attain mode locking, as described with reference to thesecond embodiment. The laser mode locking, on the other hand, is aphenomenon in which the Fabry-Perot mode interval becomes constant,because of the third order optical non-linearity. Therefore, it ispossible that mode locking cannot be generated, when equality ofrespective mode separation is lost because an interface between media ofdifferent optical characteristics exist within the resonator 1. Thiswill be studied by numerical analysis in the following.

When a semiconductor mode locked laser is operated freely, the spectrumwidth becomes wider as the injection current increases, and the pulsetime width reaches the order of several hundreds femto seconds. (Y.Nomura et al., Technical Report of IEICE (In Japanese), LQE99-8, 45(1999); Y. Nomura, et al., in Abstracts in The 6th InternationalWorkshop on Femtosecond Technology, Chiba, July 1999). In order totransmit the mode locked optical pulse through an optical fiber, it isnecessary to make narrower the spectrum, so as to minimize the wideningof the pulse time width caused by dispersion. For this purpose, it iseffective to incorporate distributed Bragg reflectors 4 a at opposingends of the resonator 1.

(Calculation)

When a Fabry-Perot resonator structure is given and a white light isintroduced from an incident end, well known periodic structures appearon the transmittance and reflectivity spectra. M. Born and E. Wolf,Principles of Optics, Fourth Edition (Pergamon Press, N.Y., 1970). Bystudying how the transmittance and reflectivity spectra differ fromthose of Fabry-Perot structure, when a divided gain section andtransparent sections as non-gain sections are inserted into theresonator to obtain a mode locked laser of long resonator, it ispossible to examine the influence of the interface of the dielectric onthe mode.

For a multilayered film structure having different index of refraction,it is possible to calculate transmittance spectrum and reflectivityspectrum, by using a characteristic matrix. M. Born and E. Wolf,Principles of Optics, Fourth Edition (Pergamon Press, New York, 1970).The problem here is the transmittance and reflectivity for thewaveguided light beam of the semiconductor laser. Therefore, in order toapply the characteristic matrix method to the waveguided light beam,effective index of refraction of the waveguide mode in the gain sectionand the transparent section should be calculated. By considering theeffective index of refraction as the index of refraction of themultilayered thin film, the characteristic matrix method is applied tothe waveguided light beam. This approach is justified, as the spatialdependency of the electric field of light can be separated into acomponent in the direction of progress and a component orthogonalthereto.

Similarly, for the DBR structure, the effective indexes of refractionfor two different types of waveguide structures are calculated bysolving the wave equation in advance, and the characteristic matrix iscalculated, regarding the structure as the multilayered stackedstructure of thin films having two different indexes of refraction.

(A. Formulation)

(A. 1 Wave Equation)

In any of the above described examples, absorption for light energy issmall, and therefore, it is assumed that the index of refraction isreal. The waveguided light satisfies the following wave equation.$\begin{matrix}{{\left\{ {{- \left( {\frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}}} \right)} + {\mu_{0}ɛ_{0}{\kappa (y)}\frac{\partial^{2}}{\partial t^{2}}}} \right\} ɛ} = 0} & (100)\end{matrix}$

In the equation (100), κ (y) represents a specific dielectric constantof the waveguide path structure. It is assumed that the light beampropagates in the z direction. For simplicity, it is assumed that thefield is homogenous in the x direction.

(A. 2 Characteristic Matrix)

When we represent the index of refraction of the thin film consisting ofa medium 2 by n₂ and the thickness by h₂, the characteristic matrix M₂thereof will be represented by the following, when the light entersvertically. $\begin{matrix}{{M_{2} = \begin{bmatrix}{\cos \quad \beta_{2}} & {{- \frac{1}{n_{2}}}\sin \quad \beta_{2}} \\{{- i}\quad n_{2}\sin \quad \beta_{2}} & {\cos \quad \beta_{2}}\end{bmatrix}}{w\quad h\quad e\quad r\quad e}} & (101) \\{\beta_{2} = {\frac{2\pi}{\lambda}n_{2}h_{2}}} & (102)\end{matrix}$

λ represents wavelength of light in vacuum.

The characteristic matrix of stacked thin films including a large numberof different media can be obtained as a product of respectivecharacteristic matrixes.

M=M ₂ ×M ₃ ×M ₄  (103)

When N pairs of the stacked structure including medium 2 and medium 3are stacked, each component of the characteristic matrix will be$\begin{matrix}{{M_{11} = {{\left\lbrack {{\cos \quad \beta_{2}\cos \quad \beta_{3}} - {\frac{n_{3}}{n_{2}}\sin \quad \beta_{2}\sin \quad \beta_{3}}} \right\rbrack {u_{N - 1}(a)}} - {U_{N - 2}(a)}}}{M_{12} = {{- {i\left\lbrack {{\frac{1}{n_{3}}\cos \quad \beta_{2}\sin \quad \beta_{3}} + {\frac{1}{n_{2}}\sin \quad \beta_{2}\cos \quad \beta_{3}}} \right\rbrack}}{u_{N - 1}(a)}}}{M_{21} = {{- {i\left\lbrack {{n_{3}\cos \quad \beta_{2}\sin \quad \beta_{3}} + {n_{2}\sin \quad \beta_{2}\cos \quad \beta_{3}}} \right\rbrack}}{u_{N - 1}(a)}}}{M_{22} = {{\left\lbrack {{\cos \quad \beta_{2}\cos \quad \beta_{3}} - {\frac{n_{2}}{n_{3}}\sin \quad \beta_{2}\sin \quad \beta_{3}}} \right\rbrack {u_{N - 1}(a)}} - {U_{N - 2}(a)}}}} & (104) \\{a = {{\cos \quad \beta_{2}\cos \quad \beta_{3}} - {\frac{1}{2}\left( {\frac{n_{2}}{n_{3}} + \frac{n_{3}}{n_{2}}} \right)\sin \quad \beta_{2}\sin \quad \beta_{3}}}} & (105)\end{matrix}$

Here, U_(N) (a) is a Chebyshev polynominal of the second kind, andtherefore, $\begin{matrix}{{U_{N}(a)} = \begin{matrix}\frac{\sin \left\lbrack {\left( {N + 1} \right)\cos^{- 1}a} \right\rbrack}{\sqrt{1 - a^{2}}} & \left( {{- 1} < a < 1} \right) \\\frac{\left( {a - \sqrt{a^{2} - 1}} \right)^{N + 1} - \left( {a + \sqrt{a^{2} - 1}} \right)^{N + 1}}{2\sqrt{a^{2} - 1}} & \left( {a < {- 1}} \right)\end{matrix}} & (106)\end{matrix}$

(A. 3 Reflectivity and Transmittance)

When the following characteristic matrix is given for an arbitraryoptical multilayered structure, $\begin{matrix}{M = \begin{bmatrix}m_{11} & m_{12} \\m_{21} & m_{22}\end{bmatrix}} & (107)\end{matrix}$

the reflectivity coefficient r and transmittance coefficient t can begiven by $\begin{matrix}{r = \frac{{\left( {m_{11} + {m_{12}n_{l}}} \right)n_{1}} - \left( {m_{21} + {m_{22}n_{l}}} \right)}{{\left( {m_{11} + {m_{12}n_{l}}} \right)n_{1}} + \left( {m_{21} + {m_{22}n_{l}}} \right)}} & (108) \\{t = \frac{2n_{1}}{{\left( {m_{11} + {m_{12}n_{l}}} \right)n_{1}} + \left( {m_{21} + {m_{22}n_{l}}} \right)}} & (109)\end{matrix}$

where, n₁ and n_(l) (the suffix of the former is a numeral 1, and thesuffix of the latter is a small l) represent indices of refraction ofsemi-infinite space in contact with opposing ends of the multilayeredstructure in question. The reflectivity R and the transmittance T arerepresented by the following. $\begin{matrix}{{R = |r|^{2}},{T = \left. \frac{n_{l}}{n_{1}} \middle| t \right|^{2}}} & (110)\end{matrix}$

(B. Result of Calculation)

(B. 1 Heterojunction Structure of the Device)

The laser assumed in calculation has an SCH structure of multiplequantum well consisting of InGaAsP/InP heterojunction structure,lattice-matched with an InP substrate (Table 3: A gain section), andoscillates at a wavelength of 1.55 μm band. The relation between thearsenic composition and the index of refraction described in G. P.Agrawal and N. K. Dutta, Long-Wavelength Semiconductor Lasers, (VanNortstrand Reinhold Company, New York, 1986) is used. The transparentsection is realized by replacing the multiple quantum well in the gainsection structure with InGaAsP having the As composition of 0.6431(Table 3: C transparent section). The DBR is provided by periodicallyarranging the transparent section (C) and the structure adapted to havehigh index of refraction layer (Table 3: B transparent section) alongthe axial direction of the resonator. Although it is dependent on thefabrication process, it may be easier to use the B transparent sectionof Table 3, as the transparent section. This case is assumed in thefollowing calculation.

The effective indices of refraction of the sections A, B and C in thestructure shown in the table calculated by solving the equation (100)are as follows: A=3.4284, B=3.4281, and C=3.4259.

(B. 2 Reflectivity-Transmittance Spectra of a Resonator Having “GainSection (A) of 1000 μm”)

FIG. 13A and 13B represent the reflectivity spectrum and transmittancespectrum of the resonator having “gain section with the length of 1000μm (Table 3: A gain section)”. Near the wavelength of 1.55 μm, all themodes have approximately the same reflectivity and transmittance,respectively, representing typical spectra of the Fabry-Perot resonator.As the reflectivity spectrum and the transmittance spectrum satisfy therelation of R+T=1, only the reflectivity spectrum will be discussed inthe following.

(B. 3 Reflectivity Spectrum of a Resonator Consisting of “TransparentSection (B) of 250 μm+Gain Section (A) of 500 μm+Transparent Section (B)of 250 μm”)

FIG. 14 shows the reflectivity spectrum of the resonator consisting of“transparent section (B) of 250 μm+gain section (A) of 500μm+transparent section (B) of 250 μm”. Except for the differenceresulting from slight difference of optical path length, the spectrum isbasically the same as the reflectivity spectrum shown in FIG. 13A.

TABLE 3 Stacked Structure Considered for Calculation p- InGaAsP Adaptedto InGaAs p- have high p-InGaAsP quantum InGaAsP A Gain InP refractivep-InP light well barrier n-InGaAsP n-InP section clad index barrierconfinement 8 layers 7 layers light confinement clad layer ∞ 0 100 80 610 80 ∞ thickness (nm) Arsenic 0 0.6431  0 0.6431 1 0.6431 0.6431 0composition p · InGaAsP B transparent section adapted to have adapted tohave high p-InP high refractive n-InP InGaAsP n · InP refractive indexclad index barrier light confinement clad layer thickness (nm) ∞ 30 100278 ∞ Arsenic composition 0 0.6431  0 0.6431 0 C transparent sectionInGaAsP adapted to have high p-InP light n-InP refractive index cladconfinement clad layer thickness (nm) ∞ 278 ∞ Arsenic composition 00.6431 0

(B. 4 Reflectivity Spectrum of a Resonator Consisting of “TransparentSection (B) of 270 μm+Gain Section (A) of 500 μm+Transparent Section (B)of 250 μm”)

FIG. 15 shows the reflectivity spectrum of a resonator consisting of“transparent section (B) of 270 μm+gain section (A) of 500μm+transparent section of 250 μm (B)”. This corresponds to an example inwhich one transparent section is made longer by 20 μm than the exampleof FIG. 14. When a laser is fabricated, if cutting is done by cleavage,instability of the resonator length to this extent is unavoidable. FromFIG. 15, it can be seen that the minimum value of reflectivity, andtherefore maximum value of transmittance fluctuate by about 10%. Suchfluctuation is of course undesirable. If the magnitude of fluctuation iswithin 10%, however, it is sufficiently smaller than the variation inreflectivity caused by DBR structure. Therefore, such fluctuation is notconsidered a problem.

(B. 5 Reflectivity Spectrum of a Resonator Consisting of “DBR Consistingof Periodic Structure of Transparent Section (B) and Transparent Section(C)+Transparent Section (B) of 250 μm+Gain Section (A) of 500μm+Transparent Section (B) of 250 μm”)

FIGS. 16 to 20 represent reflectivity spectra of a resonator consistingof “DBR consisting of periodic structure of transparent section (B) andtransparent section (C)+transparent section (B) of 250 μm+gain section(A) of 500 μm+transparent section (B) of 250 μm”. The number of pairs inrespective DBRs are (A) 40, (B) 500, (C) 1000, (D) 2000 and (E) 5000.FIG. 21 shows the reflectivity spectrum where the number of pairs in theDBR is 2000, with the length of one transparent section elongated to 270μm.

From FIGS. 16 to 20, it can be understood that the reflectivity spectrumhas a peak of which height increases gradually and the width decreases,as the number of pairs increases. Up to the pair number of 2000 pairs(see FIG. 19), the Fabry-Perot mode can be distinguished. When thenumber of pairs is 5000 pairs (see FIG. 20), the Fabry-Perot mode cannotbe observed in the so-called stop band. Therefore, 2000 pairs is almostthe upper limit of the number of pairs in which the Fabry-Perot mode canbe distinguished. Here, it is expected that three Fabry-Perot modesoscillates simultaneously. Assuming that these oscillate in mode-lockedmanner, and the pulse width will be 4.8 ps, assuming the TL (TransferLimit) (δtδf≦0.36).

In FIG. 21, three modes have high reflectivity as in the case of FIG.19, while the height of the reflectivity is not symmetrical. This isbecause the lengths of the transparent sections are asymmetrical. Byinjecting carriers to the transparent section, the following phasecompensation is possible. $\begin{matrix}{{\delta\varphi} = \frac{2\pi \quad d\quad \delta \quad \overset{\_}{n}}{\lambda}} & (h)\end{matrix}$

where d represents the length of the section B, and

δ{overscore (n)}  (i)

represents variation in the index of refraction resulting from carrierinjection. If the following variation of the index of refraction ispossible by the carrier injection, perfect phase compensation isrealized.

δ{overscore (n)}=6.2×10⁻³  (j)

According to the present invention, as supported by the numericalanalysis, even by a laser apparatus having such a resonator length thatcauses FMML oscillation if the entire inner portion of the resonator isused as the gain section, AMML oscillation can be obtained.Particularly, when the resonator length is long and the entire innerportion of the resonator is used as the gain section, oscillation tendsto be in FMML, and therefore, the effect to obtain the AMML state by theapplication of the present invention is significant. By attaining theAMML state, high optical pulse can be obtained, which greatlycontributes to realize ultra high speed/ultra large capacity opticaltransmission.

Although the present invention has been described and illustrated indetail, it is clearly understood that the same is by way of illustrationand example only and is not to be taken by way of limitation, the spiritand scope of the present invention being limited only by the terms ofthe appended claims.

What is claimed is:
 1. A laser apparatus comprising: a resonator havingan optical path length and including: a gain section in which apopulation inversion is produced, two non-gain sections in which gainwith respect to laser light produced in said laser apparatus is notpositive, said two non-gain sections sandwiching said gain section alongthe optical path length, and two reflection mirrors sandwiching saidgain section and said two non-gain sections, wherein said gain sectionhas an optical path length approximately one half the optical pathlength of said resonator and is centrally located within said resonator;and gain section excitation means for exciting said gain section.
 2. Thelaser apparatus according to claim 1, wherein said gain section includesa semiconductor material and each of said non-gain sections includes adielectric material.
 3. The laser apparatus according to claim 1,wherein said gain section includes a semiconductor material and each ofsaid non-gain sections includes a semiconductor material.
 4. The laserapparatus according to claim 3, comprising respective non-gain sectionelectrodes on said two non-gain sections for controlling refractiveindex of said two non-gain sections.
 5. The laser apparatus according toclaim 1, wherein at least one of said reflection mirrors is adistributed Bragg reflector.
 6. The laser apparatus according to claim5, further comprising a distributed Bragg reflector electrode on saiddistributed Bragg reflector for changing a reflectivity spectrum of saiddistributed Bragg reflector.
 7. The laser apparatus according to claim3, wherein said gain excitation means includes an electrode at said gainsection.
 8. The laser apparatus according to claim 7, comprisingrespective non-gain section electrodes at said two non-gain sections forcontrolling refractive index of said two non-gain sections.
 9. The laserapparatus according to claim 8, wherein at least one of said reflectionmirrors is a distributed Bragg reflector.
 10. The laser apparatusaccording to claim 9, further comprising a distributed Bragg reflectorelectrode at said distributed Bragg reflector for changing areflectivity spectrum of said distributed Bragg reflector.
 11. A laserapparatus comprising: a resonator having an optical path length andincluding: a gain section in which a population inversion is produced,two non-gain sections in which gain with respect to laser light producedin said laser apparatus is not positive, said two non-gain sectionssandwiching said gain section along the optical path length, and tworeflection mirrors sandwiching said gain section and said two non-gainsections, wherein said gain section has an optical path length in arange from eight percent to fifty percent of the optical path length ofsaid resonator and is centrally located within said resonator; and gainsection excitation means for exciting said gain section.
 12. The laserapparatus according to claim 11, wherein said gain section includes asemiconductor material and each of said non-gain sections includes adielectric material.
 13. The laser apparatus according to claim 11,wherein said gain section includes a semiconductor material and each ofsaid non-gain sections includes a semiconductor material.
 14. The laserapparatus according to claim 13, comprising respective non-gain sectionelectrodes at said two non-gain sections for controlling refractiveindex of said two non-gain sections.
 15. The laser apparatus accordingto claim 13, wherein said gain excitation means includes an electrode atsaid gain section.
 16. The laser apparatus according to claim 15,comprising respective non-gain section electrodes at said two non-gainsections for controlling refractive index of said two non-gain sections.17. The laser apparatus according to claim 16, wherein at least one ofsaid reflection mirrors is a distributed Bragg reflector.
 18. The laserapparatus according to claim 17, further comprising a distributed Braggreflector electrode at said distributed Bragg reflector for changing areflectivity spectrum of said distributed Bragg reflector.
 19. The laserapparatus according to claim 11, wherein at least one of said reflectionmirrors is a distributed Bragg reflector.
 20. The laser apparatusaccording to claim 19, further comprising a distributed Bragg reflectorelectrode at said distributed Bragg reflector for changing areflectivity spectrum of said distributed Bragg reflector.